Exponential decay for a nonlinear axially moving viscoelastic string

This paper is concerned with the stabilization of a nonlinear axially moving viscoelastic string. First, we show under suitable conditions on the initial data that solutions exist globally by using the potential well method. Then, we prove that the produced damping by the viscoelastic term is sufficient to guarantee an exponential decay of solutions. This is established by imposing weaker conditions than the commonly used ones on the relaxation function and by utilizing an appropriate boundary controller.